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Full text of "Analytical Mechanics"

148
ANALYTICAL MECHANICS
ILLUSTRATIVE EXAMPLES.
1. Find the center of mass of an octant of a homogeneous sphere, (a) Suppose the bounding surfaces to be
x2 + 2/2 + z2 = a2, 3 = 0, y = 0, and z = 0. Then the limits of integration are
x = 0 and x = a,
2/ = 0 and y = \/a2  #2,
25 = 0 and z = Va2  x2  y2.
Therefore
nVa2-a-2         Va2-a;2-y2 I     xdxdy dz y. -_      -       ______JQ___________
nC   dx dy dz J o
and by symmetry y = z = -
o
(b) Suppose the equations of the bounding surfaces to be given in spherical coordinates, then we have
r = a, 0 =    , 0 = 0, and 0 =    -
The limits of integration are
r = 0    and   r = a, 0 = 0    and   0 = ^,
and   0 = ~ -
FIG. 81.
,.,....._ xiiciciuic
fl            TT
j^2 j^2 j^ar3sin2 0 cos0dr d6 dcf>
_          .   ..         .
[a; = r sin ^ cos0] .
3a
; 8 *